The optimal control problem of slew maneuver of a spacecraft (as solid body) from arbitrary initial attitude into required final angular position is considered and solved. Designing and construction of optimal control of spacecraft rotation is based on the quaternion variables and Pontryagin's maximum principle. The case when the conditions of transversality have crucial significance is investigated. The minimized functional takes into account energy of rotation and duration of maneuver. Using the necessary conditions of optimality in the form of maximum principle and the quaternion method for solving the control problems of spacecraft motion, an analytical solution to the formulated problem was obtained. On basis of the conditions of transversality as the necessary conditions of optimality, main properties, laws and key characteristics (parameters, constants, integrals of motion) of optimal solution of control problem, including the maximal rotation energy for optimum motion and duration of maneuver are determined. The formalized equations and the calculated expressions and dependences for synthesis of the optimal program of reorientation are obtained. The estimation of influence of constraint and narrow-mindedness of the controlling moment onto character of optimal motion and onto indexes of quality of control is given. It is shown that the accepted criterion of optimality guarantees motion of a spacecraft with energy of rotation which does not exceed the required value. For spherically symmetric spacecraft and dynamically symmetric spacecraft, the complete solution of reorientation problem is presented in the closed form. The example and results of mathematical simulation of spacecraft motion under optimal control, showing a practical realizability of the developed method for control of spatial orientation of a spacecraft, are given.